Statistical theory of correlations in random packings of hard particles

A random packing of hard particles represents a fundamental model for granular matter. Despite its importance, analytical modeling of random packings remains difficult due to the existence of strong correlations which preclude the development of a simple theory. Here, we take inspiration from liquid theories for the $n$-particle angular correlation function to develop a formalism of random packings of hard particles from the bottom-up. A progressive expansion into a shell of particles converges in the large layer limit under a Kirkwood-like approximation of higher-order correlations. We apply the formalism to hard disks and predict the density of two-dimensional random close packing (RCP), $\phi_{\rm rcp} = 0.85\pm0.01$, and random loose packing (RLP), $\phi_{\rm rlp} = 0.67\pm0.01$. Our theory also predicts a phase diagram and angular correlation functions that are in good agreement with experimental and numerical data.